The present invention relates to signal data processing and in particular to efficient resampling of non-uniformly sampled data for medical imaging.
The Fourier Transform (FT) is one of the most important signal processing techniques in use today. In particular, it finds a number of uses in medical imaging, including reconstruction of MRI images and Fourier reconstruction of CT images. In MRI applications, the FT is used to convert the acquired data into an image. The quality and accuracy of the image is of utmost importance in medical examinations.
The Fast Fourier Transform (FFT), is an efficient implementation of the FT, which can only be utilized on data that is uniformly sampled in a transform domain. In addition to the FFT, there are other signal and image processing techniques that require the input data to be sampled to a specific grid, for example, backprojection reconstruction in CT or MRI imaging and diffraction tomography.
In many real-world situations, the data is not uniformly sampled. In spiral MRI, for example, the non-uniform sampling is due to variations in magnetic gradients and timing circuits. Typically, allowing non-uniform sampling can significantly shorten MRI data acquisition time.
In MRI imaging, data is acquired into a signal space called a k-space, which is the Fourier transform space of the image. An image is usually reconstructed from the k-space by applying an FFT to the data in the k-space. A major difference between MRI methods is the order in which data is acquired into the k-space. For example, in spiral MRI, data is acquired along a spiral trajectory in a two dimensional k-space, while in spin echo MRI, data is acquired along individual rows in the k-space. In 3D MRI, the k-space is three dimensional.
The commonly used solution to non-uniformly spaced data points is to interpolate the data points onto a uniformly spaced grid. One method of interpolation (referred to herein as the GRD method) was originally devised for radio astronomy by W. N. Brouw in xe2x80x9cAperture Synthesisxe2x80x9d, B. Adler, S. Fernbach and M. Rotenberg, editors, Methods in Computational Physics, Vol. 14, pp. 131-175, Academic press, New York, 1975. This method was introduced into medical imaging by O""Sullivan in xe2x80x9cA Fast Sinc Function Gridding Algorithm for Fourier Inversionxe2x80x9d, IEEE Trans. Med. Imaging, MI-4:200-207, 1985 and by Jackson et. al in xe2x80x9cSelection of A Convolution Function for Fourier Inversion using Griddingxe2x80x9d, in IEEE Trans. Med. Imaging, MI-10:473-478, 1991, and further elaborated by Meyer et. al in xe2x80x9cFast Spiral Coronary Artery Imagingxe2x80x9d, Magn. Reson. Med., 28:202-213, 1992, the disclosures of which are incorporated herein by reference.
In a conventional gridding method, resampling of data (in k-space) is applied as follows:
(a) Pre-compensate the sampled data with the inverse of the sampled data density, to compensate for the varying density of sampling in k-space.
(b) Convolute the data with a Kaiser-Bessel window function.
(c) Re-sample onto a uniformly spaced Cartesian grid.
(d) Perform an FFT on the redistributed set of data points to get an image.
(e) Post-compensate the transformed data to remove apodization of the convolution kernel by dividing the image data by the transform of the Kaiser-Bessel window function.
A Kaiser-Bessel convolution is used, rather than a sinc convolution, to reduce the computational complexity. The preferred convolution kernel is zero outside a certain window size, so that each resampled data point will be interpolated only from a small number of data points, in its vicinity, rather than using all or most of the data points in the set, as would be required in a sinc based interpolation.
The step of post-compensation is required to correct for a so-called roll-off effect induced by the transform of the convolution window. Even after the post compensation, there is generally a degradation of image quality towards the edges of the generated image. Two types of effects are generally visible: xe2x80x9ccuppingxe2x80x9d and xe2x80x9cwingsxe2x80x9d. Cupping is where the intensity profile is lower (or higher) at the center of the image than at its ends and wings is where there is an overrun of the signal beyond the ends of the image carrying portion of the image. xe2x80x9cDensity Compensation Functions for Spiral MRIxe2x80x9d, by R. Hoge et al., in MRM 38:117-128 (1997), the disclosure of which is incorporated herein by reference, provides in FIG. 5D thereof a graphic example of both degradation effects for a Jackson type gridding algorithm. Such degradation is undesirable in medical images that are used for diagnosis. One solution is to interpolate onto a 2Nxc3x972N grid, rather than onto an Nxc3x97N grid (oversampling). The result is then post-compensated and only the central Nxc3x97N portion of the post-compensated result is persevered. Most of the artifacts are outside this central portion. However, this technique increases the number of points for the FFT, by a factor of four, which considerably increases the complexity of the computation.
In the performance of real time MRI imaging, e.g., imaging of the heart and imaging fluid dissipation in tissues, the number of computational steps allowed between sequential images should be kept to a minimum. Typically, the pre-compensation, convolution and resampling are performed by multiplying the column-stacked data by a suitable, pre-calculated, matrix of coefficients. In some cases, the pre-compensation is applied separately. The post-compensation requires an element-by-element multiplication by a pre-calculated matrix. The number of required calculations is very important in medical imaging since the size of MRI images can be as large as 1,024xc3x971,024 or more.
An article titled xe2x80x9cComparison of Interpolating Methods for Image Resamplingxe2x80x9d by J. Parker et al, IEEE trans. on Medical Imaging March 1983, states that the choice of an interpolating function for resampling depends upon the task being performed. When verisimilar images are desired, this article suggests cubic B-spline interpolation. When additional processing of the images is to be performed the article suggests high-resolution cubic spline interpolation.
Another source of image degradation is noise. Substantially every sampled data includes noise, such as, thermal noise from the source of the sampled data and noise due to the apparatus used in acquiring the data. In resampling, the noise in the original data is passed over to the resampled data.
In medical images, certain types of noise are found to be more tolerable to the human observer than others. For example, as mentioned in the above referenced article, noise which is correlated with an image is much more noticeable than noise which is uncorrelated with the image. In viewing medical images, it is commonly desired to receive images which have a clear appearance, i.e. a low level of local noise, even if the received images are less accurate, i.e., have a higher bias.
U.S. Pat. No. 4,982,162 to Zakhor et al. and an article from the same author titled xe2x80x9cOptimal sampling and Reconstruction of MRI Signals Resulting from Sinusoidal Gradientsxe2x80x9d, IEEE transactions on signal processing, September 1991, the disclosures of which are incorporated herein by reference, describe derivation of a one-dimensional least square estimator matrix for generating an image from non-uniform sampled data, based on estimation theory. The estimator requires a matrix inversion which is time consuming. The time required for matrix inversion is a function of the number of sampled data points, which in two dimensional images is on the order of tens of thousands.
One object of some preferred embodiments of the invention is to provide a method of uniform resampling of data in one or more dimensions, which method is computationally efficient.
An object of some preferred embodiments of the invention is to provide a method of uniform resampling that does not introduce significant errors into an image reconstructed from the resampled data. Additionally, by not introducing errors, an extra step of correcting for those errors is not required. Alternatively or additionally, the variance of the errors is reduced, so that the errors are more uniformly distributed over the image.
An object of some preferred embodiments of the invention is to provide a method of data resampling which reduces the effect of the noise from the original data, on the resampled data.
An object of some preferred embodiments of the invention is to provide a method of data resampling which is computationally stable.
It is an object of some preferred embodiments of the invention to provide a method of data resampling, preferably for generating an image, which allows a user to control a tradeoff between an SNR of the image and a bias of the image.
In accordance with a preferred embodiment of the invention, data is resampled by determining interpolation coefficients that could be used to convert from uniformly spaced data to non-uniformly spaced data. These coefficients are inverted to determine interpolation coefficients that convert non-uniformly spaced data into uniformly spaced data. In a preferred embodiment of the invention, the coefficients are not inverted as one unit, which would be very time consuming. Rather, when inverting a particular coefficient, only coefficients in a (mathematical) locality of that coefficient are used to calculate the proper inversion.
One aspect of some preferred embodiments of the present invention relates to a method of resampling non-uniformly sampled data onto a uniform grid. The method is introduced by first examining the inverse problem, i.e., how to obtain a non-uniformly sampled data vector from a uniformly sampled data vector. The solution to the inverse problem is then inverted, to yield a solution to the forward problem. An equation Ax=b describes the relationship between a non-uniformly sampled data vector b, a uniformly sampled data vector x and a coefficient matrix A (e.g., sinc coefficients) which converts the uniformly sampled vector into a non-uniformly sampled vector. In a MRI example of the inverse problem, data from a uniformly sampled k-space is placed in vector x and multiplication by matrix xe2x80x9cAxe2x80x9d yields vector b of data corresponding to a non-uniformly sampled k-space. In the forward problem, data from a non-uniformly sampled k-space is stacked into vector b and multiplied by some matrix to obtain a uniformly sampled vector x.
In accordance with a preferred embodiment of the invention, the vector x is determined from vector b, by multiplying vector b with an inverse of coefficient matrix A, A#, i.e., x=A#b. In some cases, A is a non-square matrix, so a true inverse cannot be defined. Preferably, A# is a pseudo-inverse of the coefficient matrix A, preferably a Moore-Penrose pseudo-inverse. In some cases, there may be a plurality of possible pseudo-inverse matrices. Preferably, a matrix that minimizes an error criterion is used. Preferably, the error criteria used is |Axxe2x88x92b|2. Preferably, the matrix which generates a minimum norm for x, is used.
In a preferred embodiment of the invention, A# is found using a singular value decomposition algorithm (SVD), most preferably a rank-truncated SVD, for example as described in W. H. Press, et. al, xe2x80x9cNumerical Recipes in Cxe2x80x9d, Cambridge University Press, Cambridge, 2nd Edition, 1992, for the real case, and in S. L. Marple Jr., xe2x80x9cDigital Spectral Analysis,xe2x80x9d Prentice-Hall, Inc., Englewood Cliffs, N.J., 1987, for the complex case. The disclosures of both these publications are incorporated herein by reference.
However, determining an SVD pseudo-inverse may be computationally expensive, especially where the matrix A is very large.
In accordance with a preferred embodiment of the invention, use is made of the fact that, during resampling, each resampled data point can be interpolated to an acceptable accuracy by using only a subset of the data points in its vicinity, rather than by using all the data points in the set. Rather than invert the entire matrix A as a single unit, an approximate inversion of matrix A is performed by decomposing the inversion problem into component problems. Each component problem relates a subset of points from vector b with a subset of points from vector x, using a subset of the coefficients from matrix A. Preferably, each subset of matrix A includes only coefficients which relate points in a limited portion of k-space. For example, an i""th sub-problem concerns relating the i""th point of vector x (xi) with a small number of points from vector b, preferably using a subset of coefficients from matrix A. This subset is represented herein by an interpolation sub-matrix Ai. In a preferred embodiment of the invention, only sub-matrices Ai are inverted (typically pseudo-inverted), giving matrices Ai#, rather than inverting the whole matrix A. In addition to relating xi, the i""th sub-problem preferably also relates additional points from vector x, preferably points in the k-space neighborhood of xi. However, generally, Ai is chosen to have the smallest possible size which will still yield a good approximation for xi when matrix Ai is inverted. In some preferred embodiments of the invention, a component problem may be used to relate more than one point in vector x to points in vector b, at a desirable accuracy.
The conversion of vector b into vector x is preferably performed by multiplying vector b by a composite inverted matrix . The composite inverted matrix is preferably assembled from the inverted sub-matrices Ai#. For each pseudo-inverted sub matrix Ai#, one row corresponds to the interpolation coefficients required for its respective xi. The i""th row of matrix  is initialized to zero and subsequently the values from the corresponding matrix Ai# are copied into the appropriate locations in the i""th row of . The appropriate locations correspond to the locations of the points in b which were selected for the i""th problem. Ultimately, matrix  comprises mostly zeros, thus, multiplying by  requires fewer steps than multiplying by A#. In addition,  takes up less storage space, as only the non-zero elements need be stored.
The size of each of the sub-matrices Ai is dependent on the amount of original data used to interpolate each resampled data point. In accordance with one preferred embodiment of the invention, the sub-matrix size is determined so that points from a constant radius (in k-space) around the resampled point are used for the resampling. Alternatively, the sub-matrix size is determined responsive to the quality of data acquisition for that region of k-space. Thus, each resampled point may be interpolated from a different number of original data points.
In a preferred embodiment of the invention, not all the points in the locality of the resampled point are used for interpolation. Preferably, substantially only a minimum number of points necessary to comply with the sampling theorem are used for the interpolation. Preferably, the points are selected to be evenly distributed in the locality (the size of the sub matrix). Preferably, a few extra points will be used to obtain a certain degree of oversampling. Preferably, the oversampling is utilized to compensate for the selected points not being evenly distributed in the locality. Additionally or alternatively, the oversampling is utilized to compensate for the signal not being completely band-limited and/or to compensate for the effects of noise. In one preferred embodiment of the invention, some of the points are ignored. Alternatively, they are averaged. In some preferred embodiments of the invention, the number of points used for resampling is a constant. Alternatively or additionally, the number of points is restricted to a maximal value. In some cases, the size of the locality may be increased to allow for a minimum number of points to be included in the interpolation. Alternatively or additionally, the size of the locality is also limited, to be within a range of sizes. In some preferred embodiments of the invention, the size of the locality is fixed.
In accordance with a preferred embodiment of the invention, provision is made for filtering the resampled data, without adding any computational steps. Preferably, this is achieved by using a matrix "PHgr"A#, rather than a matrix A#, where "PHgr", a convolution matrix, performs the effect of the filter, i.e., x=[("PHgr"A#]b. In some preferred embodiments of the invention, where  is used instead of A#, each of the sub-matrices used to assemble  is pre-multiplied by a portion of "PHgr". Preferably, the k-space coverage of each sub-matrix of , which coverage determines the size of the matrix, is selected to be larger than the impulse response of the filter used, so that the use of portions of the filter, rather than a whole filter, does not substantially affect the filter performance. Alternatively, the k-space coverage is selected to be at least larger than an important portion of the impulse response. The important portion is preferably defined as a portion which contains most of the energy of the impulse response.
It should be appreciated that the above described methods are better suited than prior art methods to handle an arbitrary order of acquisition of data in MRI, since a pre-compensation for local density is not required. An extreme example of an arbitrary data acquisition is a stochastic k-space trajectory, as described for example in xe2x80x9cFrequency Resolved Single Shot MR Images Using Stochastic k-Space Trajectoriesxe2x80x9d, by K. Scheffler and J. Hennig, in Magn. Reson. Med., vol. 35, pp. 569-576, 1996, the disclosure of which is incorporated herein by reference. In addition, some types of pre-compensation are tailored to the use of particular k-space trajectories and/or rates of travel along the trajectory, i.e., the gradients applied. By doing away with pre-compensation, in accordance with some preferred embodiments of the invention, there is less of a need to tailor the resampling technique to the imaging technique.
An aspect of some preferred embodiments of the present invention relates to a density pre-compensation matrix. Preferably, the density pre-compensation matrix includes elements which are negative. Preferably, the density pre-compensation matrix is calculated by finding a diagonal pre-compensation matrix which when it pre-multiplies matrix A and is pre-multiplied by a matrix A#, yields the identity matrix (or at least a minimum error with the identity matrix). In a preferred embodiment of the invention, the use of such a calculated pre-compensation matrix allows an independence from a k-space trajectory velocity profile. One example of a previously problematic k-space trajectory is a stochastic trajectory. Another example is variable density spiral MRI imaging, as described for example in J. R. Liao and J. M. Pauly and T. J. Brosnan and N. J. Pelc, xe2x80x9cReduction of Motion Artifact in Cine MRI Using Variable-Density Spiral Trajectoriesxe2x80x9d, Magn. Reson. Med., vol. 37, pp. 569-575, 1997, the disclosure of which is incorporated herein by reference. Still another example is an MRI square-like spiral trajectory, described for example in A. Macovski and C. H. Meyer, xe2x80x9cA novel fast-scanning system,xe2x80x9d Works in Progress, Fifth Annual Meeting of the Society of Magnetic Resonance in Medicine, 1986, pp. 156-157, the disclosure of which is incorporated herein by reference.
An aspect of some preferred embodiments of the invention relates to real-time CT image reconstruction. In faster CT machines, data is acquired using a fan beam and the data must be rebinned to form a parallel beam. In addition, the k-space acquired by most CT imagers is non-uniformly sampled. In order to apply some Fourier reconstruction methods, the k-space should preferably be uniformly resampled.
In some preferred embodiments of the invention, where the coefficient matrix is inverted using sub-matrices, use is made of the fact that data points in vector x are dependent mainly on a small number of points in vector b. In one preferred embodiment of the invention, data points are resampled as soon as the required corresponding data points are acquired into vector b and without waiting for a data vector to be acquired. Preferably, this use is applied to process partial updates of the values in k-space.
In a preferred embodiment of the invention, where a CT image is continuously updated, for example as described in PCT applications PCT/IL98/00074, filed on Feb. 12, 1998, titled xe2x80x9cReal Time Dynamic Image Reconstructionxe2x80x9d and PCT/IL98/00075, filed on Feb. 12, 1998, titled xe2x80x9cOn Line Image Reconstruction in Helical CT Scannersxe2x80x9d and in U.S. Pat. No. 5,524,130, issued on Jun. 4, 1996, the above-described method can be applied to increase the rate and/or quality of image generation. In these references, a CT image is created using information from a previous and mostly overlapping image, in addition to a small amount of new information. In a preferred embodiment of the invention, data, from a plurality of projections, are arranged in a vector b. A vector x is resampled from the vector b, using a matrix , where each sub matrix is preferably compatible with the size of a projection. When a new projection is inserted into the vector b, it is not necessary to resample the entire vector b into a new vector x. Rather, a new vector x can be reconstructed by xnew=xold+(bnewxe2x88x92bold). Preferably, this equation is limited so that it is only applied to the new projection and to data points which are affected by the new data, such as being in their vicinity.
In accordance with an aspect of some preferred embodiments of the invention, resampling is performed using estimation theory. Resampled values of the data at the resampled points are determined so as to minimize an error criterion, such as the minimum variance unbiased (MVU) criterion, the maximum a-priori (MAP) criterion or the maximum likelihood (ML) criterion. The resampled values represent estimations of the sampled signal at the resampled points or estimations of a function of the signal, e.g., estimations of filtered values of the signal. In the following text, the term xe2x80x9csampled signalsxe2x80x9d refers to the pure signal without any accompanying noise. The term xe2x80x9csampled dataxe2x80x9d, however, refers to the samples which include noise.
In some preferred embodiments of the present invention, each resampled value is estimated based on a sub-group of the sampled data points. Preferably, each resampled value is estimated based on a sub-group of the sampled points in its locality. Preferably, values of a sub-group of resampled points are estimated together based on a sub-group of the sampled points in a locality of the resampled points.
In some preferred embodiments of the present invention, the resampled data is estimated using linear estimation methods. Alternatively or additionally, non-linear estimation methods are used.
One aspect of some preferred embodiments of the invention relates to resampling using statistical information of the sampled and/or resampled data. Preferably, an optimal estimator for translating the sampled data onto the points of the desired resampled data is determined, given the statistical information. In a preferred embodiment of the present invention, the estimator comprises a function of the statistical information and of an interpolation matrix suitable for resampling in an opposite direction, i.e., from the resampled points to the sampled points.
In some preferred embodiments of the present invention, the sampled data is formed of a signal component and a noise component, and the statistical information relates to the noise component. Alternatively or additionally, the statistical information relates to the signal component and/or to the resampled data. Further alternatively or additionally, the statistical information relates to the correlation between the noise component and the signal component and/or the resampled data. In some preferred embodiments of the present invention, the statistical information comprises one or more moments of the noise components and/or of the signal components of the sampled data. Alternatively or additionally, the statistical information comprises joint moments of the signal and noise components, and/or joint moments of the resampled data with the noise and/or signal components. Preferably, the one or more moments comprise first and/or second order moments. Alternatively or additionally, the statistical information comprises a probability density function (PDF). In a preferred embodiment of the present invention, the statistical information comprises a signal-to-noise-ratio (SNR) of the sampled data. Preferably, a single SNR value is determined for substantially all the sampled points in a single acquisition session. Alternatively, different SNR values are used for different resampled points or for different sampled points.
In some preferred embodiments of the present invention, the statistical information represents characteristics of an acquisition apparatus. Preferably, the statistical information is determined at a calibration stage of the apparatus and is provided with the apparatus by a manufacturer of the apparatus. Further preferably, the statistical information is periodically updated based on manufacturer updates and/or based on cumulative information from acquisition procedures performed by the apparatus. Alternatively or additionally, the statistical information is selected from a predetermined table based on the nature of the specific data being sampled. Further alternatively or additionally, the statistical information is estimated from the sampled data of a current acquisition session.
In a preferred embodiment of the present invention, other interpolation coefficients are used in the resampling process in addition to or instead of the interpolation matrix. Alternatively or additionally, the estimator is dependent on other functions which take into account the values of the sampled data points and/or the positions of the data points.
An aspect of some preferred embodiments of the invention relates to setting parameters of a resampling process of sampled data, responsive to one or more attributes associated with the sampled data. Preferably, the parameters comprise parameters of an estimator. Preferably, the one or more attributes are determined during and/or immediately before or after the acquisition session in which the sampled data is acquired. Alternatively or additionally, the one or more attributes are determined from the sampled data.
In a preferred embodiment of the present invention, the sampled data is used for medical imaging of a patient. The one or more attributes preferably comprise the identity of the organ being imaged, the geometry of the organ, the age of the patient being imaged and/or other information about the patient. Alternatively or additionally, the one or more attributes comprise at least one characteristic of the acquisition process. The at least one characteristic of the acquisition process preferably includes an attribute of the sequence type, an acquisition sequence parameter and/or an attribute of an acquisition apparatus. Further alternatively or additionally, the one or more attributes comprise an attribute which represents the type of imaging sequence performed, e.g., the shape of the k-space.
In a preferred embodiment of the present invention, the one or more resampling parameters comprise a parameter which controls a tradeoff between a noise level and a bias in a reconstructed image.
In some preferred embodiments of the present invention, the one or more resampling parameters are used in selecting a resampling estimator. Preferably, based on the one or more attributes, an estimator is chosen from a predetermined list of estimators. Alternatively or additionally, the one or more attributes are used to choose an optimality criterion according to which the estimator is chosen. Further alternatively or additionally, the estimator used in resampling is a function of one or more of the resampling parameters. Preferably, the resampling parameters comprise statistical information regarding the sampled or resampled data.
In a preferred embodiment of the present invention, the one or more attributes are used to retrieve from a predetermined list, statistical information regarding the signal and/or noise components of the sampled data and/or a-priori statistical information regarding the resampled data. The predetermined list is preferably prepared based on a plurality of previous acquisition procedures.
In some preferred embodiments of the present invention, the determined one or more attributes associated with the sampled data are used in choosing and/or adjusting a filter for use in a filtering stage before and/or after the resampling.
An aspect of some preferred embodiments of the invention relates to resampling using an estimator which has one or more adjustable resampling parameters. A physician sets the adjustable parameters based on the particular application of the resampled data and/or according to trial and error. Alternatively or additionally, the adjustable parameters are determined iteratively so as to maximize a given function of the resampled data, preferably of a function of an image created from the resampled data.
Preferably, the adjustable parameters receive values along a continuum. Further preferably, the effect of the values of the parameters on the resampling is continuous. Alternatively or additionally, the adjustable parameters receive a discrete number of values.
In some preferred embodiments of the present invention, when the resampled data is used for generating an image the adjustable parameters are chosen from a first group of values. When the resampled data is used for other applications the adjustable parameters are chosen from other values.
There is therefore provided in accordance with a preferred embodiment of the invention, a method of resampling medical imaging data from a first spatial distribution of data points onto a second spatial distribution of data points, including determining a matrix of reverse interpolation coefficients for resampling data from the second spatial distribution onto the first spatial distribution, inverting a matrix based on the reverse interpolation matrix to determine forward resampling coefficients for resampling data from the first spatial distribution to the second spatial distribution, and resampling data from the first spatial distribution onto the second spatial distribution using the forward resampling coefficients.
Preferably, the matrix based on the reverse interpolation matrix includes inverting the reverse interpolation matrix.
Alternatively or additionally, the matrix based on the reverse interpolation matrix includes the sum of the reverse interpolation matrix multiplied by its Hermitian conjugate and a parameter matrix.
Preferably, the parameter matrix includes a diagonal matrix.
Preferably, all the non-zero elements of the parameter matrix are substantially equal.
Preferably, the parameter matrix includes a correlation matrix.
Preferably, the resampling coefficients include interpolation coefficients.
Alternatively or additionally, the resampling coefficients include estimation coefficients.
Preferably, determining the reverse interpolation matrix includes determining a real matrix.
Preferably, the second spatial distribution includes a uniform spatial distribution.
Preferably, the first spatial distribution includes a non-uniform spatial distribution.
Preferably, the second spatial distribution includes a radial spatial distribution or a Cartesian spatial distribution.
Preferably, the medical imaging data includes Magnetic Resonance k-space data.
Alternatively or additionally, the medical imaging data includes Magnetic Resonance imaging data.
Further alternatively or additionally, the medical imaging data includes Magnetic Resonance spectroscopy data.
Alternatively or additionally, the medical imaging data includes CT k-space data.
Further alternatively or additionally, the medical imaging data includes CT projection data, converted from fan-beam to parallel beam.
Further alternatively or additionally, the medical imaging data includes diffraction tomography k-space data.
Preferably, inverting includes calculating a pseudo-inverse matrix.
Alternatively or additionally, inverting includes inverting using rank truncated SVD (Singular Value Decomposition).
Preferably, the determining is performed locally on the first and second spatial distributions.
Preferably, the inverting is performed locally on the first and second spatial distributions.
Preferably, determining a matrix of resampling coefficients includes selecting {overscore (M)} points from the second spatial distribution and {overscore (N)} points from the first spatial distribution, for each of the localities.
Preferably, the {overscore (M)} points are selected from a first region surrounding a point xi.
Preferably, {overscore (M)} is dependent on the locality.
Preferably, the first region is circular, having a first radius dependent on the locality.
Alternatively or additionally, the first region is non-circular and/or rectangular.
Preferably, the {overscore (N)} points are selected from a second region surrounding a point xi.
Preferably, {overscore (N)} is dependent on the locality.
Preferably, the second region is circular, having a second radius dependent on the locality.
Alternatively or additionally, the second region is non-circular and/or rectangular.
Preferably, the resampling includes generating an inversion matrix and each row is created from an inversion at a locality.
Preferably, copying resampling coefficients resulting from the inversion into a zeroed matrix row of the inversion matrix, which row corresponds to point xi.
Preferably, the determining is performed using a grid different from the second spatial distribution.
Preferably, the different grid has a greater extent than the second spatial distribution.
Alternatively or additionally, the different grid has a different spacing than the second spatial distribution.
Further alternatively or additionally, the different grid has a larger and/or smaller spacing than the second spatial distribution.
Preferably, the resampling includes pre-multiplying a matrix including the forward interpolation coefficients, by a filter.
Preferably, the filter has a FIR (Finite Impulse Response) and the FIR is smaller than an extent of the locality.
Preferably, the filter has an impulse response having most of its energy concentrated within an area smaller than an extent of the locality.
Preferably, the resampling includes resampling spatial data having dimensionality greater than one.
Preferably, reconstructing an image from the resampled data by applying an FFT (Fast Fourier Transform) to the data.
There is further provided in accordance with a preferred embodiment of the invention, a method of resampling including, providing data in a first spatial distribution of data points, providing a second spatial distribution of data points, and resampling data from the first spatial distribution onto the second spatial distribution, without generating artifacts in the data, which artifacts could be corrected by pixel-by-pixel multiplying an image reconstructed from the resampled data, by a pre-determined post-compensation matrix, the resampling being performed by multiplying the data by a single matrix.
Preferably, the single matrix is a sparse matrix in which each row includes at least 20% zero elements.
Further preferably, the single matrix is a sparse matrix in which each row includes at least 50% zero elements.
Further preferably, the single matrix is a sparse matrix in which each row includes at least 80% zero elements.
Preferably, the second spatial distribution includes a uniform spatial distribution.
Preferably, the first spatial distribution includes a non-uniform spatial distribution.
There is further provided in accordance with a preferred embodiment of the invention, a method of resampling including providing data in a first spatial distribution of data points, providing a second spatial distribution of data points, pre-multiplying the data by a diagonal density pre-compensation matrix which includes at least one element having a negative value, and resampling the data from the first spatial distribution onto the second spatial distribution.
Preferably, the diagonal pre-compensation matrix includes both positive and negative elements.
Preferably, the method includes reconstructing an image from the resampled data by applying an FT (Fourier Transform) to the data.
Preferably, the method includes pixel-by-pixel multiplying the reconstructed image by a pre-determined post-compensation matrix.
There is further provided in accordance with a preferred embodiment of the invention, a method of determining a diagonal density pre-compensation matrix, including providing a first spatial distribution of data points, providing a second spatial distribution of data points, determining a first interpolation matrix for resampling data from the first spatial distribution to the second spatial distribution, determining a second interpolation matrix for resampling data from the second spatial distribution to the first spatial distribution, and determining a diagonal pre-compensation matrix which minimizes an error between an identity matrix and the multiplication of the first and second interpolation matrices.
Preferably, the diagonal pre-compensation matrix includes elements having negative values.
Further preferably, the diagonal pre-compensation matrix includes both positive and negative elements.
Preferably, the first interpolation matrix is generated by multiplying two or more matrices.
Preferably, the determining a diagonal pre-compensation matrix includes generating a set of equations.
Preferably, generating a set of equations includes generating a matrix equation, which equation includes a multiplication relationship between a plurality of matrices.
Preferably, the plurality of matrices includes a backwards interpolation matrix, a diagonal pre-compensation matrix, an interpolation coefficient matrix and a convolution matrix.
Preferably, the method includes for each element on the diagonal of the diagonal matrix, selecting only a portion of the backwards interpolation matrix.
Preferably, the portion corresponds to portions of the backwards interpolation matrix which generate a non-zero value when multiplied by the diagonal element.
Alternatively or additionally, the portion corresponds to portions of the backwards interpolation matrix which correspond to uniformly sampled data points within a region in k-space surrounding a data point represented by the diagonal element.
Preferably, the region is circular or rectangular.
Preferably, selecting includes selecting only some of the parts of the backwards interpolation matrix which correspond to data points within the region.
Preferably, the portions include rows and/or columns.
Preferably, the method includes for each element on the diagonal of the diagonal matrix, selecting only a portion of the convolution matrix.
Preferably, the portion of the convolution matrix corresponds to uniformly sampled data points within a second region in k-space surrounding a data point represented by the diagonal element.
Preferably, the second region is circular or rectangular.
Preferably, selecting includes selecting only some of the parts of the convolution matrix which correspond to data points within the second region.
There is further provided in accordance with a preferred embodiment of the invention, a method of resampling data organized in a first spatial distribution of sampled data points onto a second spatial distribution of resampled data points, including, obtaining statistical information pertaining to the sampled data or the resampled data, and estimating the values of the resampled data points responsive to the obtained statistical information and to the sampled data.
Preferably, obtaining the statistical information includes acquiring data containing substantially only noise and determining the statistical information therefrom.
Alternatively or additionally, obtaining the statistical information includes guessing the statistical information.
Further alternatively or additionally, obtaining the statistical information includes estimating the statistical information from the sampled data.
Further alternatively or additionally, obtaining the statistical information includes estimating the statistical information using one or more sets of previously acquired sampled data.
Further alternatively or additionally, obtaining the statistical information includes retrieving the statistical information from a table.
Preferably, retrieving the statistical information from a table includes retrieving the information responsive to one or more attributes of the data.
Preferably, the one or more attributes include an identity of an organ represented by the data and/or a geometry of an imaged area and/or a determined noise level.
Preferably, obtaining the statistical information includes determining the statistical information based on a characteristic of an apparatus used to sample the sampled data.
Preferably, determining the statistical information includes determining based on a rate of sampling and a bandwidth of the sampled data.
Preferably, obtaining the statistical information includes obtaining statistical information regarding the resampled data and/or the sampled data.
Preferably, obtaining the statistical information includes obtaining statistical information regarding a noise component and/or a signal component of the sampled data.
Alternatively or additionally, the statistical information includes a signal-to-noise-ratio.
Further alternatively or additionally, the statistical information includes a probability density function (PDF) of the sampled data.
Preferably, the statistical information includes one or more statistical moments.
Alternatively or additionally, the statistical information includes correlation information.
Preferably, the statistical information includes auto-correlation information.
Preferably, estimating the values of the resampled data points includes determining for each of the resampled data points an estimator which is a function of the statistical information, and calculating the value of the resampled data point by applying the estimator to at least some of the sampled data points.
Preferably, the estimator includes a Bayesian estimator.
Alternatively or additionally, the estimator includes a non-linear estimator.
Alternatively, the estimator includes a linear estimator.
Preferably, the estimator includes a mean of a posterior PDF of the resampled data.
Preferably, the estimator is a function of a set of interpolation coefficients.
Preferably, the estimator includes an optimal linear Bayesian mean square error (MSE) estimator.
Further preferably, the estimator includes the equation x=xcexcx+(AHCNxe2x88x921A+Cxxe2x88x921)xe2x88x921AHCNxe2x88x921(bxe2x88x92Axcexcx), in which x represents the resampled data, b represents the sampled data, A represents the set of interpolation coefficients, and CN, Cx, and xcexcx represent the statistical information.
Alternatively or additionally, the estimator includes a function of a product of an interpolation matrix multiplied by its Hermitian conjugate and by a matrix which represents the statistical information.
Preferably, the estimator includes a function of a matrix inverse of the product of the interpolation matrix multiplied by its Hermitian conjugate and by the matrix which represents the statistical information.
Preferably, the estimator is of the form x=(AHCNxe2x88x921A)xe2x88x921AHCNxe2x88x921b, wherein x represents the resampled data, b represents the sampled data, A represents the set of interpolation coefficients, and CN represents the statistical information.
Preferably, the matrix which represents the statistical information includes a correlation matrix of a noise component of the sampled data.
Preferably, the interpolation matrix includes a real matrix.
Preferably, the set of interpolation coefficients includes interpolation coefficients suitable for resampling the second spatial distribution of data points onto the first spatial distribution of data points.
Preferably, the estimator minimizes an estimation error criterion.
Preferably, the estimation error criterion includes a root mean square error criterion.
Preferably, estimating the values of the resampled data points includes estimating responsive to a sub-group of the sampled data points.
Preferably, the sub-group of sampled data points of a resampled data point includes sampled data points in a region surrounding the resampled data point.
Preferably, estimating the values of the resampled data points includes estimating the values of a signal component of the data at the resampled data points.
Alternatively or additionally, estimating the values of the resampled data points includes estimating the values of a function of the data at the resampled data points.
Preferably, estimating the values of the resampled data points includes estimating filtered values of the data at the resampled data points.
Preferably, estimating includes solving a set of equations of the form   E  ⁢      {                            (                                    x              i                        -                                          ∑                                  m                  =                  1                                K                            ⁢                              xe2x80x83                            ⁢                                                                    y                    im                                    ⁡                                      (                                                                  b                        m                                            +                                              v                        m                                                              )                                                  ⁢                                  (                                                            b                      k                                        +                                          v                      k                                                        )                                                              }                =        0            ,      
in which xi represent the resampled data, bi represent a signal component of the sampled data, v represents a noise component of the sampled data, and yim represent the estimator.
Preferably, the sampled data includes medical imaging data.
Further preferably, the medical imaging data includes Magnetic Resonance k-space data.
Alternatively or additionally, the medical imaging data includes CT imaging data.
Preferably, the estimating includes estimating spatial data having dimensionality greater than one.
There is further provided in accordance with a preferred embodiment of the invention, a method of resampling medical imaging data organized in a first spatial distribution of sampled data points onto a second spatial distribution of resampled data points, including determining at least one attribute of the source of the data, and estimating the values of the resampled data points from the sampled data points responsive to the determined attribute.
Preferably, the at least one attribute includes an attribute of the object being imaged and/or an identity of a body part being imaged.
Alternatively or additionally, the at least one attribute includes an age group of a patient being imaged.
Preferably, the at least one attribute includes an attribute of an acquisition process of the sampled data.
Preferably, the at least one attribute includes an attribute of an acquisition sequence type.
Alternatively or additionally, the at least one attribute includes an attribute of an acquisition sequence parameter.
Alternatively or additionally, the at least one attribute includes an attribute of an acquisition apparatus.
Alternatively or additionally, estimating the values of the resampled data points includes selecting an estimator responsive to the determined at least one attribute.
Preferably, estimating the values of the resampled data points includes parametrically adjusting an estimator responsive to the determined at least one attribute.
Preferably, adjusting the estimator includes selecting parameters of the estimator from a look up table, responsive to the at least one attribute.
There is further provided in accordance with a preferred embodiment of the invention, a method of resampling data organized in a first spatial distribution of sampled data points onto a second spatial distribution of resampled data points, including providing an estimator which depends on a parameter independent of the first and second spatial distributions, setting a value of the parameter, and applying the estimator to the sampled data points to receive values for the resampled data points.
Preferably, providing the estimator includes selecting an estimator which minimizes an error criterion.
Preferably, the error criterion includes a weighted error criterion, the weights representing an importance of the accuracy of the values of the resampled data points.
Preferably, providing the estimator includes selecting an estimator according to availability of statistical information.
Preferably, setting the value of the parameter includes selecting a value from a discrete number of possible values or from a continuum of possible values.
Preferably, setting the value of the parameter includes setting the value responsive to an attribute of the sampled data.
Preferably, providing the estimator includes providing an estimator which is a function of an interpolation matrix.
Alternatively or additionally, providing the estimator includes providing an estimator which is a function of a sum of the interpolation matrix multiplied by its Hermitian conjugate and a parameter matrix.
Preferably, the parameter matrix includes a diagonal matrix and/or a correlation matrix.
Preferably, applying the estimator includes inverting the sum of the product of the interpolation matrix multiplied by its Hermitian conjugate and of the parameter matrix.
Preferably, inverting includes inverting using SVD.
Preferably, providing the estimator includes providing an estimator which is a function of a sum of a first parameter matrix and the product of the interpolation matrix multiplied by its Hermitian conjugate and by a second parameter matrix.
Preferably, the interpolation matrix includes a real matrix.
Preferably, applying the estimator includes applying the estimator to subsets of the sampled data.
Alternatively or additionally, applying the estimator includes applying the estimator so as to receive the values of the resampled data points responsive to respective surrounding sampled points.
Preferably, setting the value of the parameter includes setting the parameter separately for each of the subsets of the sampled data.
There is further provided in accordance with a preferred embodiment of the invention, a method of estimating a set of MRI-related values, including acquiring a set of MRI values which are related to the estimated MRI-related values through a linear model determining an association matrix which defines a linear association between a sub-group of the estimated values and a sub-group of the sampled values, and estimating the set of MRI related values by applying an estimating matrix to the sampled set of values, the estimating matrix being a function of the sum of a product matrix and a first parameter matrix, the product matrix being a product of the association matrix multiplied by its Hermitian conjugate and by a second parameter matrix.
Preferably, the first parameter matrix includes a diagonal matrix.
Alternatively or additionally, the second parameter matrix includes a unit matrix.
Preferably, the estimating matrix includes a function of an inverse of the sum of the product matrix and the first parameter matrix.
Preferably, the second parameter matrix includes a correlation matrix representing noise added to the sampled set of MRI values during acquisition.
Preferably, the association matrix includes a matrix of interpolation coefficients.
Preferably, applying the estimating matrix includes applying to a subset of the set of values.
There is further provided in accordance with a preferred embodiment of the invention, a method of resampling data organized in a first spatial distribution of sampled data points onto a second spatial distribution of resampled data points, including acquiring sampled data, and applying an optimal linear Bayesian mean square error estimator to the sampled data points so as to receive values for the resampled data points.
Preferably, applying the estimator includes setting arbitrarily at least one matrix of statistical data required by the estimator.
Preferably, setting arbitrarily includes assigning a diagonal matrix value.
There is further provided in accordance with a preferred embodiment of the invention, a method of resampling data organized in a first spatial distribution of sampled data points onto a second spatial distribution of resampled data points, including applying a first estimator to a first sub-group of the sampled data points to receive values for a first sub-group of the resampled data points, and applying a second estimator to a second sub-group of the sampled data points to receive values for a second sub-group of resampled data points.
Preferably, the second estimator is different from the first estimator.
There is further provided in accordance with a preferred embodiment of the invention, apparatus for resampling data organized in a first spatial distribution of sampled data points onto a second spatial distribution of resampled data points, including a medical imaging receiver which acquires the sampled data, an input interface which receives statistical information regarding the sampled data or the resampled data, and a processor which estimates the values of the resampled data points responsive to the statistical information and to the sampled data.
Preferably, the apparatus includes a memory which stores a look up table of statistical information suitable for various types of sampled data.
Preferably, the processor applies an optimal linear Bayesian mean square error estimator to the sampled data
There is further provided in accordance with a preferred embodiment of the invention, apparatus for imaging, including a medical imaging receiver which samples a plurality of sampled data points, and a processor which resamples the sampled data points by applying an estimator to the sampled data points, and converts the resampled data points into an image, wherein the estimator is dependent on a parameter unrelated to the sampled data.
Preferably, the receiver includes an MRI receiver.